David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
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Logica Universalis 3 (2):303-332 (2009)
The central aim of this paper is to present a Boolean algebraic approach to the classical Aristotelian Relations of Opposition, namely Contradiction and (Sub)contrariety, and to provide a 3D visualisation of those relations based on the geometrical properties of Platonic and Archimedean solids. In the first part we start from the standard Generalized Quantifier analysis of expressions for comparative quantification to build the Comparative Quantifier Algebra CQA. The underlying scalar structure allows us to define the Aristotelian relations in Boolean terms and to propose a 3D visualisation by transforming a cube into an octahedron. In part two, the architecture of the CQA is shown to carry over, both to the classical quantifiers of Predicate Calculus and to the modal operators—which are given a Generalized Quantifier style re-interpretation. In this way we provide an algebraic foundation for Blanché’s Aristotelian hexagon as well as a 3D alternative to his 2D star-like visualisation. In a final part, a richer scalar structure is argued to underly the realm of Modality, thus generalizing the 3D algebra with eight (2 3 ) operators to a 4D algebra with sixteen (2 4 ) operators. The visual representation of the latter structure involves a transformation of the hypercube to a rhombic dodecahedron. The resulting 3D visualisation allows a straightforward embedding, not only of the classical Blanché star of Aristotelian relations or the paracomplete and paraconsistent stars of Béziau (Log Investig 10, 218–232, 2003) but also of three additional isomorphic Aristotelian constellations.
|Keywords||logical square logical hexagon aristotelian relations of opposition contradiction contrariety Boolean algebra modal logic polyhedra|
|Categories||categorize this paper)|
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References found in this work BETA
Laurence Horn (1989). A Natural History of Negation. University of Chicago Press.
Barbara H. Partee, Alice ter Meulen & Robert E. Wall (1992). Mathematical Methods in Linguistics. Journal of Symbolic Logic 57 (1):271-272.
Lloyd Humberstone (2005). Modality. In Frank Jackson & Michael Smith (eds.), The Oxford Handbook of Contemporary Philosophy. Oxford University Press
Citations of this work BETA
Alessio Moretti (2015). Was Lewis Carroll an Amazing Oppositional Geometer? History and Philosophy of Logic 35 (4):383-409.
Hans Smessaert & Lorenz Demey (2014). Logical Geometries and Information in the Square of Oppositions. Journal of Logic, Language and Information 23 (4):527-565.
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